To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .

To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services
Please confirm that you accept the terms of use.

In his Mathematical Foundations of Quantum Mechanics, John von Neumann presented a proof that the quantum statistics cannot be recovered from probability distributions over “hidden” deterministic states that assign definite premeasurement values to all physical quantities. Von Neumann's proof has been dismissed as “silly” by John Bell and by David Mermin, who writes:

Many generations of graduate students who might have been tempted to try to construct hidden-variables theories were beaten into submission by the claim that von Neumann, 1932, had proved that it could not be done. A few years later (see Jammer, 1974, p. 273) Grete Hermann, 1935, pointed out a glaring deficiency in the argument, but she seems to have been entirely ignored. Everybody continued to cite the von Neumann proof. A third of a century passed before John Bell, 1966, rediscovered the fact that von Neumann's nohidden-variables proof was based on an assumption that can only be described as silly—so silly, in fact, that one is led to wonder whether the proof was ever studied by either the students or those who appealed to it to rescue them from speculative adventures.

Discussions of quantum-computational algorithms in the literature refer to various features of quantum mechanics as the source of the exponential speed-up relative to classical algorithms: superposition and entanglement, the fact that the state space of n bits is a space of 2n states while the state space of n qubits is a space of 2n dimensions, the possibility of computing all values of a function in a single computational step by “quantum parallelism,” or the possibility of an efficient implementation of the discrete quantum Fourier transform. Here I propose a different answer to the question posed in the title, in terms of the difference between classical logic and quantum logic, i.e., the difference between the Boolean classical event structure and the non-Boolean quantum event structure. In a nutshell, the ultimate source of the speed-up is the difference between a classical disjunction, which is true (or false) in virtue of the truth values of the disjuncts, and a quantum disjunction, which can be true (or false) even if none of the disjuncts is either true or false.

In the following, I will discuss the information-processing in Deutsch's XOR algorithm (the first genuinely quantum algorithm) and related period-finding quantum algorithms (Simon's algorithm and Shor's factorization algorithm). It is well known that these algorithms can be formulated as solutions to a hidden-subgroup problem. Here the salient features of the information-processing are presented from the perspective of the way in which the algorithms exploit the non-Boolean logic represented by the projective geometry (the subspace structure) of Hilbert space.

Recommend this

Email your librarian or administrator to recommend adding this to your organisation's collection.